Algebraic models of cyclic $k$-gonal curves

TL;DR

This paper provides explicit algebraic equations for tame cyclic k-gonal curves, extending previous work to non-prime k, and elucidates the automorphism group's action on these curves.

Contribution

It introduces a general method for deriving algebraic models of tame cyclic k-gonal curves for any integer k, broadening the scope beyond prime k cases.

Findings

Explicit algebraic equations for tame cyclic k-gonal curves are derived.

The action of the normalizer of the automorphism group is characterized.

The work generalizes previous results from prime to composite k.

Abstract

In this paper, we describe explicit algebraic equations of tame cyclic $k$-gonal curves, where $k \geq 2$ is an integer, reflecting the action of the normalizer of a tame cyclic $k$-gonal automorphism. For $k$ a prime integer, this was previously done by A. Wootton.